Klein Group Research Articles

Low ML Decoding Complexity STBCs via Codes Over the Klein Group

In this paper, we give a new framework for constructing low ML decoding complexity space-time block codes (STBCs) using codes over the Klein group <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> . Almost all known low ML decoding complexity STBCs can be obtained via this approach. New full-diversity STBCs with low ML decoding complexity and cubic shaping property are constructed, via codes over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> , for number of transmit antennas <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> =2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ≥ 1, and rates <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> >; 1 complex symbols per channel use. When <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> , the new STBCs are information-lossless as well. The new class of STBCs have the least known ML decoding complexity among all the codes available in the literature for a large set of ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> ) pairs.

The braided monoidal structures on the category of vector spaces graded by the Klein group

AbstractLet k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].

The torsion-free part of the Ziegler spectrum of the Klein four group

We will describe the torsion-free part of the Ziegler spectrum, both the points and the topology, over the integral group ring of the Klein group. For instance we will show that the Cantor–Bendixson rank of this space is equal to 3.

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.

Duality Functors for Triple Vector Bundles

We calculate the group of dualization operations for triple vector bundles, showing that it has order 96 and not 72 as given in Mackenzie's original treatment. The group is a nonsplit extension of S4 by the Klein group. Dualization operations are interpreted as functors on appropriate categories and are said to be equal if they are naturally isomorphic. The method set out here will be applied in a subsequent paper to the case of n-fold vector bundles.

Covering Spaces of Arithmetic 3-Orbifolds

Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has infinite virtual first Betti number and G has super-exponential subgroup growth. Our first result, which forms the basis for the entire paper, is that G is commensurable with a lattice containing the Klein group of order 4. Our second result is that if G has a finite index subgroup with first Betti number at least 4, then G is large. This is known to hold for many arithmetic lattices. In particular, we show that it is always the case when G contains A 4 , S 4 or A 5 . Our third main result is that the Lubotzky-Sarnak conjecture and the geometrisation conjecture together imply that any arithmetic Kleinian group G is large. We also give a new ‘elementary’ proof that arithmetic Kleinian groups do not have the congruence subgroup property, which avoids the use of the Golod-Shafarevich inequality.

On the Zudilin-Rivoal theorem

We propose a new method for proving the Zudilin-Rivoal theorem stating, in particular, that the sequence of values of the Dirichlet beta function at even natural points contains infinitely many irrational values. For polylogarithms, we use Hermite—Pade approximations of the first type, invariant with respect to the Klein group. Quantitative additions to this theorem are obtained.

Generic polynomials for transitive subgroups of the group S 4 over fields of characteristic 2

Generic polynomials for the symmetric group S4, the Klein group V4, the cyclic group of order 4, the dihedral group D4, and the alternating group A4 over fields of characteristic 2 are described explicitly. Bibliography: 5 titles.

Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168

There is a family of seventh-degree polynomials $H$ whose memberspossess the symmetries of a simple group of order $168$. This grouphas an elegant action on the complex projective plane. Developingsome of the action's rich algebraic and geometric properties rewardsus with a special map that also realizes the $168$-fold symmetry.The map's dynamics provides the main tool in an algorithm thatsolves certain 'heptic' equations in $H$.

Modularity of hypertetrahedral representations

Let F be a number field, G F its absolute Galois group, and ρ : G F → GL 4 ( C ) an irreducible continuous Galois representation. Let G ¯ denote the projective image of ρ in PGL 4 ( C ) . We say that ρ is hypertetrahedral if G ¯ is an extension of A 4 by the Klein group V 4 . In this case, we show that ρ is modular, i.e., ρ corresponds to an automorphic representation π of GL 4 ( A F ) such that their L-functions are equal. This gives new examples of irreducible 4-dimensional monomial representations which are modular, but are not induced from normal extensions and are not essentially self-dual. To cite this article: K. Martin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Symmetry Preservation in the Evolution of the Genetic Code

The standard genetic code is found to exhibit an exact symmetry under a finite group of order 4 known in mathematics as the Klein group. The same symmetry is also present in almost all non-standard codes, mitochondrial as well as nuclear. Analysis of the phylogenetic tree for the evolution of the mitochondrial codes reveals that all changes along the main line of evolution preserve this symmetry, with a tendency towards symmetry enhancement. In the side branches of the evolutionary tree, the majority of changes also respect the symmetry. The few exceptional cases where it is broken correspond to reassignments that appear to be unstable or incomplete. Since the Klein group emerges naturally from the symplectic model for the prebiotic evolution that has led to the standard code, we interpret these results as lending support to the hypothesis that this symmetry has been selected during the evolution of the genetic code, not only before but also after establishment of the standard code.

Gradings of Matrix Algebras by the Klein Group

We classify the C 2 × C 2-gradings on M 2(k), k an arbitrary field. If char(k) ≠ 2 our approach relies on the duality between gradings and actions for finite abelian groups, and if k is algebraically closed we find precisely one isomorphism type of grading which is not isomorphic to a grading with all the matrix units being homogeneous elements. If char(k) = 2 we use a computational approach and we find that any C 2 × C 2-grading is induced by a C 2-grading.

Highly symmetric cellular automata and their symmetry-breaking patterns

A highly symmetric cellular automaton (in one dimension) is defined by two properties: (i) its local rule is a completely symmetric function of the n cells of its surrounding and (ii) it is invariant with respect to a regular, Abelian group acting on its M-letter alphabet, which can then be identified with this group. The motivation for studying such systems is provided by the unsolved problem of partial symmetry breaking in spin models with such an Abelian symmetry. It is shown that the symmetries (i) and (ii) do not contradict each other if and only if n and M have no common factor. As examples, the four-letter alphabets, which may be identified either with the cyclic group C(4) or with the Klein group K(4)≅ S(2)⊗ S(2) are studied for the standard n=3 surrounding. It is shown that these automata show complicated patterns of broken symmetries. These give information on the corresponding spin models, the chiral 4-state clock model and the (symmetric and general) Ashkin–Teller models, respectively.

Journal Reviews

Covington, Coline. ‘Learning how to forget – repeating the past as a defence against the present’.Grotstein, James. ‘Bion, the Pariah of O’.Hinshelwood, R. D. ‘The elusive concept of “internal objects” (1934–1943): its role in the formation of the Klein Group’.Lippmann, Paul. ‘On dreams and interpersonal psychoanalysis’.Renik, Owen. ‘Conscious and unconscious use of the self’.Schaverien, Joy. ‘Men who leave too soon: reflections on the erotic transference and countertransference’.Williams, Gianna, ‘Reflections on some dynamics of eating disorders: “no entry” defences and foreign bodies’.Slavin, Malcolm &amp; Kriegman, D. ‘Why the analyst needs to change: towards a theory of conflict, negotiation, and mutual influence in the therapeutic process’.

Wildness implies undecidability for lattices over group rings

Let G be a finite group. A Z [G]-lattice is a finitely generated Z-torsionfree module over the group ring Z [G]. There is a general conjecture concerning classes of modules over sufficiently recursive rings, and linking wildness and undecidability. Given a finite group G, Z [G] is sufficiently recursive, and our aim here is just to investigate this conjecture for Z [G]-lattices. In this setting, the conjecture says thatif and only ifIn particular, we wish to deal here with the direction from the left to the right, so the one assuring that wildness implies undecidability. Of course, before beginning the analysis of this problem, one should agree upon a sharp definition of wildness for lattices. But, for our purposes, one might alternatively accept as a starting point a general classification of wild Z [G]-lattices when G is a finite p-group for some prime p, based on the isomorphism type of G. This is due to several authors and can be found, for instance, in [3]. It says that, when p is a prime and G is a finite p-group, thenif and only if.More precisely, the representation type of Z [G]-lattices is finite when G is cyclic of order ≤ p2, tame domestic when G is the Klein group [1], tame non-domestic when G is cyclic of order 8 [11].So our claim might be stated as follows.

Hausdorff Dimension of Limit Sets for Spherical CR Manifolds

LetM2n+1(n⩾1) be a compact, spherical CR manifold. SupposeM2n+1is its universal cover andΦ:M2n+1→S2n+1is on injective CR developing map, whereS2n+1is the standard unit sphere in the complex (n+1)-spaceCn+1, thenM2n+1is of the quotient formΩ/Λ, whereΩis a simply connected open set inS2n+1, andΛis a complex Klein group acting onΩproperly discontinuously. In this paper, we show that if the CR Yamabe invariant ofM2n+1is positive, then the Carnot Hausdorff dimension of the limit set ofΛis bounded above byn·s(M2n+1), wheres(M2n+1)⩽1 and is a CR invariant. The method that we adopt is analysis of the CR invariant Laplacian. We also explain the geometric origin of this question.

The group of fuzzy homeomorphisms

In this paper we study some relations between the group of fuzzy homeomorphisms of a fuzzy topological space X and the group of permutations of the ground set X. In contrast with the results in general topology, it is proved that the subgroups generated by a cycle can be represented as the group of fuzzy homeomorphisms for some fuzzy topology on X and the group of even permutations can be represented as the group of fuzzy homeomorphisms for some fuzzy topology on X. Also it is proved that the Klein group can be represented as the group of fuzzy homeomorphisms for some fuzzy topology.

Op amp relocation and the complementary transformation

It is shown that the problem of synthesizing RC oscillators and notch filters can be reduced to a group theoretic problem of finding isomorphisms. The required group is identified as the Klein group. From this it is also shown that there is a sharp upper bound on the number of new circuits that can be generated from the knowledge of a parent circuit. This bound is independent of the number of branches and internal nodes

Upper bound on the number of equivalent oscillator‐notch filter circuits: A group theoretic approach

AbstractIt is shown that the problem of synthesizing RC oscillators and notch filters can be reduced to a group theoretic problem of finding isomorphisms. the required group is identified as the Klein group. From this it is also shown that there is a sharp upper bound on the number of new circuits that can be generated from the knowledge of a parent circuit. This bound is independent of the number of branches and internal nodes.

On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4

It is shown that there exists a resolvablen2 by 4 orthogonal array which is invariant under the Klein 4-groupK4 for all positive integersn congruent to 0 modulo 4 except possibly forn ∈ {12, 24, 156, 348}.